Fractals of Generalized F-Hutchinson Operator in b-Metric Spaces

Definition 1. Let X be a nonempty set and let b ≥ 1 be a given real number. A function $$ is said to be a b-metric if, for any x, y, z ∈ X, the following conditions hold:

•  

  b₁ d(x, y) = 0 if and only if x = y,

•  

  b₂ d(x, y) = d(y, x),

•  

  b₃ d(x, y) ≤ b(d(x, z) + d(z, y)).

The pair (X, d) is called b-metric space with parameter b ≥ 1.

If b = 1, then b-metric space is metric spaces. But the converse does not hold in general [20, 21, 25].

Example 2 (see [31].)Let (X, d) be a metric space, and ρ(x, y) = (d(x, y)) ^p, where p > 1 is a real number. Then ρ is b-metric with b = 2^p−1.

Obviously conditions (b₁) and (b₂) of above definition are satisfied. If 1 < p < ∞, then the convexity of the function f(x) = x^p(x > 0) implies

and hence (a + b)^p ≤ 2^p−1(a^p + b^p) holds. Thus, for each x, y, z ∈ X we obtain So condition (b₃) of the above definition is satisfied and ρ is b-metric.

If $$ (set of real numbers) and d(x, y) = |x − y| is the usual metric, then ρ(x, y) = (x − y) ² is b-metric on $$ with b = 2 but is not a metric on $$.

Definition 3 (see [24].)Let (X, d) be b-metric space. Then a subset C⊆X is called

• (i)

  closed if and only if, for each sequence {x_n} in C which converges to an element x, we have x ∈ C (i.e., $$),

• (ii)

  compact if and only if for every sequence of elements of C there exists a subsequence that converges to an element of C,

• (iii)

  bounded if and only if δ(C)≔sup⁡{d(x, y) : x, y ∈ C} < ∞.

Let $$ denote the set of all nonempty compact subsets of X. For $$, let

where d(x, B) = inf⁡{d(x, y) : y ∈ B} is the distance of a point x from the set B. The mapping H is said to be the Pompeiu-Hausdorff metric induced by d. If (X, d) is a complete b-metric space, then $$ is also a complete b-metric space.

Lemma 4. Let (X, d) be b-metric space. For all $$, the following hold:

•  

  i If B⊆C, then sup_a∈A⁡d(a, C) ≤ sup_a∈A⁡d(a, B).

•  

  ii sup_x∈A∪B⁡d(x, C) = max⁡{sup_a∈A⁡d(a, C), sup_b∈B⁡d(b, C)}.

•  

  iii One has H(A ∪ B, C ∪ D) ≤ max⁡{H(A, C), H(B, D)}.

Lemma 5. Let (X, d) be b-metric space and CB(X) denotes the set of all nonempty closed and bounded subsets of X. For x, y ∈ X and A, B ∈ CB(X), the following statements hold:

• (1)

  (CB(X), H) is b-metric space.

• (2)

  d(x, B) ≤ H(A, B) for all x ∈ A.

• (3)

  One has d(x, A) ≤ b(d(x, y) + d(y, A)).

• (4)

  For h > 1 and $$, there is $$ such that $$

• (5)

  For every h > 0 and $$, there is $$ such that $$

• (6)

  For every λ > 0 and $$, there is $$ such that $$

• (7)

  For every λ > 0 and $$, there is $$ such that $$ implies H(A, B) ≤ λ.

• (8)

  d(x, A) = 0 if and only if $$

• (9)

  For {x_n}⊆X,

  $$ ()

Definition 6. Let (X, d) be b-metric space. A sequence {x_n} in X is called

•  

  i Cauchy if and only if, for ε > 0, there exists $$ such that for each n, m ≥ n(ε) one has d(x_n, x_m) < ε,

•  

  ii convergent if and only if there exists x ∈ X such that for all ε > 0 there exists $$ such that for all n ≥ n(ε) one has d(x_n, x) < ε. In this case one writes lim_n→∞⁡x_n = x.

Definition 7 (see [33].)Let (X, d) be a metric space. A self-mapping f on X is called F-contraction if, for any x, y ∈ X, there exist F ∈ Ϝ and τ > 0 such that

whenever d(fx, fy) > 0.

From (F₁) and (5), we conclude that

that is, every F-contraction mapping is contractive, and in particular, every F-contraction mapping is continuous.

Definition 8. Let (X, d) be b-metric space. A self-mapping f on X is called a generalized F-contraction if, for any x, y ∈ X, there exist F ∈ Ϝ and τ ∈ Υ such that

whenever d(fx, fy) > 0.

Theorem 9. Let (X, d) be b-metric space and let f : X → X be generalized F-contraction. Then one has the following:

•  

  1 f maps elements in $$ to elements in $$.

•  

  2 If, for any $$,

  $$ ()

•  

  then $$ is a generalized F-contraction mapping on $$.

Proof. As generalized F-contractive mapping is continuous and the image of a compact subset under f : X → X is compact, we obtain

To prove (2), let $$ with H(f(A), f(B)) ≠ ∅. Since f : X → X is a generalized F-contraction, we obtain Thus we have Also Now Strictly increasing F implies Consequently, there exists a function $$ with lim inf_t→0τ(t) > 0 for all t ≥ 0 such that Hence $$ is a generalized F-contraction.

Theorem 10. Let (X, d) be b-metric space and let {f_n : n = 1,2, …, N} be a finite family of generalized F-contraction self-mappings on X. Define $$ by

Then T is a generalized F-contraction on $$.

Proof. We demonstrate the claim for N = 2. Let f₁, f₂ : X → X be two F-contractions. Take $$ with H(T(A), T(B)) ≠ 0. From Lemma 4 (iii), it follows that

Definition 11. Let (X, d) be a metric space. A mapping $$ is said to be a Ciric type generalized F-contraction if, for F ∈ Ϝ and τ ∈ Υ such that, for any A, $$ with H(T(A), T(B)) ≠ 0, the following holds:

where

Theorem 12. Let (X, d) be b-metric space and let {f_n : n = 1,2, …, N} be a finite sequence of generalized F-contraction mappings on X. If $$ is defined by

then T is a Ciric type generalized F-contraction mapping on $$.

Proof. Using Theorem 10 with property (F₁), the result follows.

Definition 13. Let X be a complete b-metric space. If f_n : X → X, n = 1,2, …, N, are generalized F-contraction mappings, then (X; f₁, f₂, …, f_N) is called generalized F-contractive iterated function system (IFS).

Thus generalized F-contractive iterated function system consists of a complete b-metric space and finite family of generalized F-contraction mappings on X.

Definition 14. A nonempty compact set A ⊂ X is said to be an attractor of the generalized F-contractive IFS if

• (a)

  T(A) = A,

• (b)

  there is an open set U⊆X such that A⊆U and lim_n→∞⁡Tⁿ(B) = A for any compact set B⊆U, where the limit is taken with respect to the Hausdorff metric.

Theorem 15. Let (X, d) be a complete b-metric space and let {X; f_n,   n = 1,2, …, k} be a generalized F-contractive iterated function system. Then the following hold:

• (a)

  A mapping $$ defined by

  $$ ()

•  

  is Ciric type generalized F-Hutchinson operator.

• (b)

  Operator T has a unique fixed point $$; that is,

  $$ ()

• (c)

  For any initial set $$, the sequence of compact sets {A₀, T(A₀), T²(A₀), …} converges to a fixed point of T.

Proof. Part (a) follows from Theorem 12. For parts (b) and (c), we proceed as follows. Let A₀ be an arbitrary element in $$ If A₀ = T(A₀), then the proof is finished. So we assume that A₀ ≠ T(A₀). Define

for $$

We may assume that A_m ≠ A_m+1 for all $$ If not, then A_k = A_k+1 for some k implies A_k = T(A_k) and this completes the proof. Take A_m ≠ A_m+1 for all $$. From (18), we have

where In case M_T(A_m, A_m+1) = H(A_m+1, A_m+2), we have a contradiction as τ(H(A_m+1, A_m+2)) > 0. Therefore M_T(A_m, A_m+1) = H(A_m, A_m+1) and we have Thus {H(A_m+1, A_m+2)} is decreasing and hence convergent. We now show that lim_m→∞⁡H(A_m+1, A_m+2) = 0. By property of τ, there exists c > 0 with $$ such that τ(H(A_m, A_m+1)) > c for all m ≥ n₀. Note that gives lim_m→∞⁡F(H(A_m+1, A_m+2)) = −∞ which together with (F₂) implies that lim_m→∞⁡H(A_m+1, A_m+2) = 0. By (F₃), there exists h ∈ (0,1) such that Thus we have On taking limit as n → ∞ we obtain that lim_m→∞⁡m[H(A_m+1, A_m+2)] ^h = 0. Hence lim_m→∞⁡m^1/hH(A_m+1, A_m+2) = 0. There exists $$ such that m^1/hH(A_m+1, A_m+2) ≤ 1 for all m ≥ n₁ and hence H(A_m+1, A_m+2) ≤ 1/m^1/h for all m ≥ n₁. For $$ with m > n ≥ n₁, we have By the convergence of the series $$, we get H(A_n, A_m) → 0 as n, m → ∞. Therefore {A_n} is a Cauchy sequence in X. Since $$ is complete, we have A_n → U as n → ∞ for some $$

In order to show that U is the fixed point of T, we on the contrary assume that Pompeiu-Hausdorff weight assigned to U and T(U) is not zero. Now

where Now we consider the following cases:

• (1)

  If M_T(A_n, U) = H(A_n, U), then, on taking lower limit as n → ∞ in (32), we have

  $$ ()

•  

  a contradiction as lim inf_t→0τ(t) > 0 for all t ≥ 0.

• (2)

  When M_T(A_n, U) = H(A_n, A_n+1), then, by taking lower limit as n → ∞, we obtain

  $$ ()

•  

  which gives a contradiction.

• (3)

  In case M_T(A_n, U) = H(U, T(U)), we get

  $$ ()

•  

  a contradiction as τ(H(U, T(U))) > 0.

• (4)

  If M_T(A_n, U) = (H(A_n, T(U)) + H(U, A_n+1))/2b, then, on taking lower limit as n → ∞, we have

  $$ ()

•  

  a contradiction as F is strictly increasing map.

• (5)

  When M_T(A_n, U) = H(A_n+2, A_n+1), then

  $$ ()

•  

  which gives a contradiction.

• (6)

  In case M_T(A_n, U) = H(A_n+2, U), then, on taking lower limit as n → ∞ in (32), we get

  $$ ()

•  

  a contradiction.

• (7)

  Finally if M_T(A_n, U) = H(A_n+2, T(U)), then, on taking lower limit as n → ∞, we have

  $$ ()

•  

  a contradiction.

Thus, U is the fixed point of T.

To show the uniqueness of fixed point of T, assume that U and V are two fixed points of T with H(U, V) being not zero. Since T is F-contraction map, we obtain that

where that is, a contradiction as τ(H(U, V)) > 0. Thus T has a unique fixed point $$.

Remark 16. In Theorem 15, if we take $$ the collection of all singleton subsets of X, then clearly $$. Moreover, consider f_n = f for each n, where f = f_i for any i ∈ {1,2, 3, …, k}; then the mapping T becomes

With this setting we obtain the following fixed point result.

Corollary 17. Let (X, d) be a complete b-metric space and let {X : f_n,   n = 1,2, …, k} be a generalized iterated function system. Let f : X → X be a mapping defined as in Remark 16. If there exist some F ∈ Ϝ and τ ∈ Υ such that, for any $$ with d(f(x), f(y)) ≠ 0, the following holds:

where then f has a unique fixed point in X. Moreover, for any initial set x₀ ∈ X, the sequence of compact sets {x₀, fx₀, f²x₀, …} converges to a fixed point of f.

Corollary 18. Let (X, d) be a complete b-metric space and let (X; f_n,   n = 1,2, …, k) be iterated function system where each f_i for i = 1,2, …, k is a contraction self-mapping on X. Then $$ defined in Theorem 15 has a unique fixed point in $$ Furthermore, for any set $$, the sequence of compact sets {A₀, T(A₀), T²(A₀), …} converges to a fixed point of T.

Proof. It follows from Theorem 10 that if each f_i for i = 1,2, …, k is a contraction mapping on X, then the mapping $$ defined by

is contraction on $$. Using Theorem 15, the result follows.

Corollary 19. Let (X, d) be a complete b-metric space and let (X; f_n,   n = 1,2, …, k) be an iterated function system. Suppose that each f_i for i = 1,2, …, k is a mapping on X satisfying

for all x, y ∈ X, f_ix ≠ f_iy, where τ ∈ Υ. Then the mapping $$ defined in Theorem 15 has a unique fixed point in $$. Furthermore, for any set $$, the sequence of compact sets {A₀, T(A₀), T²(A₀), …} converges to a fixed point of T.

Proof. Take F(λ) = ln⁡(λ) + λ, λ > 0 in Theorem 10; then each mapping f_i for i = 1,2, …, k on X satisfies

for all x, y ∈ X, f_ix ≠ f_iy, where τ ∈ Υ. Again from Theorem 10, the mapping $$ defined by satisfies for all $$ and H(T(A), T(B)) ≠ 0. Using Theorem 15, the result follows.

Corollary 20. Let (X, d) be a complete b-metric space and let (X; f_n, n = 1,2, …, k) be iterated function system. Suppose that each f_i for i = 1,2, …, k is a mapping on X satisfying

for all x, y ∈ X, f_ix ≠ f_iy, where τ ∈ Υ. Then the mapping $$ defined in Theorem 15 has a unique fixed point in $$. Furthermore, for any set $$, the sequence of compact sets {A₀, T(A₀), T²(A₀), …} converges to a fixed point of T.

Proof. By taking F(λ) = ln⁡(λ² + λ) + λ, λ > 0, in Theorem 10, we obtain that each mapping f_i for i = 1,2, …, k on X satisfies

for all x, y ∈ X, f_ix ≠ f_iy, where τ ∈ Υ. Again it follows from Theorem 10 that the mapping $$ defined by satisfies for all $$, H(T(A), T(B)) ≠ 0. Using Theorem 15, the result follows.

Corollary 21. Let (X, d) be a complete b-metric space and let (X; f_n,   n = 1,2, …, k) be iterated function system. Suppose that each f_i for i = 1,2, …, k is a mapping on X satisfying

for all x, y ∈ X, f_ix ≠ f_iy, where τ ∈ Υ. Then the mapping $$ defined in Theorem 15 has a unique fixed point $$ Furthermore, for any set $$, the sequence of compact sets {A₀, T(A₀), T²(A₀), …} converges to a fixed point of T.

Proof. Take $$, λ > 0, in Theorem 10, and then each mapping f_i for i = 1,2, …, k on X satisfies

where τ ∈ Υ. Again it follows from Theorem 10 that the mapping $$ defined by satisfies for all $$, H(T(A), T(B)) ≠ 0. Using Theorem 15, the result follows.